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Basic idea For a long time I was looking for a numerical system that would allow to compare infinite sets. In contrast to Cantor's approach that empathizes the possibility of on-to-one correspondence between sets, I want that adding an element to even an infinite set should increase its "quantity" and removing an element should decrease. For instance, 1,2,3,4,5... should have greater quantity than 1,2,4,5... Similarly I want more dense sets having greater quantity, that is, 1,2,3,4,5... having greater quantity than 1,3,5,7... and slmaller than 1,3/2,2,5/2,3,... This way I came to the following considerations. First of all, we extend the real numbers with non-standard numbers. Each non-standard number consists of a standard part and non-standard part. The numbers whose standard part is zero we call pure non-standard. For instance, if p''' is a pure non-standard number, then '''p+1 has standard part 1'''. Now we introduce the notion of 'quantity' of a subset of real numbers '''q(S). - If the set of reals is finite, then the quantity of that set is equal to the number of its members. - The quantity of all integers we designate as \Omega=2\tau . It is a pure non-standard number. - If two sets differ by only the presence or absence of finite number of elements then the non-standard parts of their quantities are equal. - If two sets differ by only the position of finite number of elements, their quantities are equal. - For non-intersecting sets S_1 and S_2 , q(S_1\cup S_2)=q(S_1)+q(S_2) - Quantities of sets symmetric against zero are equal. - Quantities of uniformly distributed sets are proportional to their densities Given these properties, lets find the quantity of the natural numbers q(\mathbb{N}) . We know that \mathbb{Z}=\{-1,-2,-3,...\}\cup\{0\}\cup\{1,2,3,...\}=\mathbb{N^-}\cup\{0\}\cup\mathbb{N} . Now q(\{0\})=1 (it is a finite set) and q(\mathbb{N^-})=q(\mathbb{N}) . So, \Omega=2q(\mathbb{N})+1 . We designate q(\mathbb{N}) as \omega_- , so \omega_-=\frac12 \Omega-\frac12=\tau-1/2 . It is not a pure non-standard number, its standard part is -1/2 . The quantity of all non-negative integers is greater by one, so we designate it \omega_+=\omega_-+1=\tau+1/2 Here are some other examples: - The quantity of even numbers is equal to the quantity of odd numbers, is equal to \Omega/2=\tau - The quantity of the numbers of the form \frac{2n-1}2 with natural n (1/2, 3/2, 5/2,...) is \frac{\Omega}2=\tau - The quantity of positive even numbers is \tau/2 , the quantity of positive odd numbers is \tau/2-1/2 , the quantity of non-negative odd numbers is \tau/2+1/2 . - The quantity of complex integers ordered lexicographically is \Omega^2=4\tau^2 Some sets and their quantities: Definition using series Now we define that to any divergent series there corresponds a non-standard number. The standard part of that number is given by the Ramanujan's sunmmation of the series. That way we see that \operatorname{st} q(\mathbb{N})= \operatorname{st} \omega_-=\sum_{n\ge1}^{\Re}1=-1/2 The quantity of a subset of natural numbers is equivalent to the series summing up the membership function of that subset. For instance, given a set S \subseteq \mathbb{N} and a function p(n) such that p(n)=1 if n\in S and p(n)=0 otherwise, q(S)=\sum_{k=1}^\infty p(n) Exponentiation of non-standard numbers Examining the Faulhaber's formula for Ramanujan's summation one can come to the following striking insight on the exponentiation of non-standard numbers. : \operatorname{st}\omega_-^n=B_n : \operatorname{st}\omega_+^n=B^'_n Where B_n are the first Bernoulli numbers and B^'_n are the second Bernoulli numbers. Indeed, we can see that \operatorname{st}\omega_-=-1/2, \operatorname{st}\omega_+=1/2, \operatorname{st}\omega_-^2=1/6, \operatorname{st}\omega_-^3=0 etc. Given that Bernoulli numbers can be expressed through Hurwitz Zeta function, we can generalize: : \operatorname{st}\omega_-^x=-x\zeta(1-x,0) : \operatorname{st}\omega_+^x=-x\zeta(1-x,1)=-x\zeta(1-x) This allows to represent zeta function in exponential form: : (x-1)\zeta(x)= \operatorname{st}\omega_-^{1-x} or, more generally, : \operatorname{st}(\omega_-+z)^n= B_n(z) : \operatorname{st}(\tau+y)^x=-x\zeta(1-x,1/2+y) Moreover, now any series containing Bernoulli numbers can be represented as power series over non-standard numbers. There are the following relations: : \operatorname{st}\tau^x = \operatorname{st}\omega_+^x (2^{1 - x} - 1) For x>1 , : \operatorname{st}\omega_-^x = \operatorname{st}\omega_+^x From the Riemann functional equation it follows: : \operatorname{st}\omega_+^{-x}=\operatorname{st}\frac{-\omega_+^{x+1} 2^x\pi^{x+1}}{\sin(\pi x/2)\Gamma(x)(x+1)} Expression for derivative If f(x) is analytic, the following holds: : f'(x)=\operatorname{st}(f(\omega_++x)-f(\omega_-+x))=\operatorname{st} \Delta f(\omega_-+x) : f'(x)=\operatorname{st}(f(\omega_++x)-f(-\omega_++x)) : f'(x)=\operatorname{st}(f(-\omega_-+x)-f(\omega_-+x)) If f(x) is odd, : \operatorname{st}f(\omega_+)=-\operatorname{st}f(\omega_-)= \frac12 f'(0) Another consequence of Faulhaber's formula connects integral with the sum: : \operatorname{st}\sum_{k=0}^\infty f(k)=-\operatorname{st}\int_0^{\omega_-} f(x) dx Improper integrals Integrals can be transformed into sets of weighted dots using the following principle: an integral over an interval (a_i,b_i) can be replaced with a weighted dot at the center of mass of the figure under the graphic of the integrated function, with weight of the figure's area: The x coordinate of the weighted dot will be x_i=\frac{\int_{a_i}^{b_i} x f(x) dx}{\int_{a_i}^{b_i} f(x) dx} the weight is : p_i=\int_{a_i}^{b_i} f(x) dx Now, if we can represent an infinite integral with a set of weighted dots located in positive integers, the corresponding series would be : \sum_{i=1}^\infty p_i One method that uses the above principle is as follows. Given the function f(x) , : p_k=\frac{\int_k^{k+1} x f(x) \, ds}{\int_k^{k+1} f(x) \, dx} : \int_0^\infty f(x)=\int_0^{p(0)} f(x) \, dx+\sum _{k=1}^{\infty } \int_{p(k-1)}^{p(k)} f(x) \, dx Consequences If f(x) is periodic and integral over period is zero, then : \int_{-\infty}^{+\infty} f(x) dx=0 If such function is also even, then : \int_0^{+\infty} f(x) dx=0 Norm One can define the norm of non-standard numbers, by analogy with complex numbers: : \|w\|=\exp(\operatorname{st}\ln w) If so, the following holds: : \|\omega_+\|=e^{-\gamma} Distribution form Integral form of Dirac Delta function is : \delta(x)=\frac1\pi\int_0^\infty \cos(xt)dt It becomes \tau/\pi at x=0 . Thus we can interpret extended numbers as derivatives of step-functions at the point of discontinuity, with step size determining the non-standard part and limits of derivatives from right and left determine the standard part. Particularly, \delta(0)=\tau/\pi \operatorname{sign}'(0)=2\tau/\pi Standard parts of some expressions Given the above definitions, we have a lot of relations between trigonometric functions, for instance, : \operatorname{st} \cos (z\omega_-)=\operatorname{st} \cos (z\omega_+)=\frac z2 \cot \left(\frac z2\right) : \operatorname{st} \cosh (z\omega_-)=\operatorname{st} \cosh (z\omega_+)=\frac z2 \coth \left(\frac z2\right) : \operatorname{st} \cos (z\tau)=\frac z2 \csc \left(\frac z2\right) : \operatorname{st} \cosh (z\tau)=\frac{z}{2} \operatorname{csch}\left(\frac{z}{2}\right) : \operatorname{st} e^{z\omega_-}=\frac{z}{e^{z}-1} : \operatorname{st} \left(\frac{1}{\pi^2 \tau+\pi x}+\frac{1}{\pi^2 \tau-\pi x}\right)=(\sec x)^2 : \operatorname{st}\ln (\omega_-+z)=\psi(z) Particularly, : \operatorname{st}\ln \omega_+=-\gamma : \operatorname{st}\frac1{\pi }\ln \left(\frac{\omega _-+\frac{z}{\pi }}{\omega _+-\frac{z}{\pi }}\right)=-\cot (z) : \operatorname{st} \frac1\pi\ln \left(\frac{\tau +\frac{z}{\pi }}{\tau -\frac{z}{\pi }}\right)=\tan (z) : \operatorname{st}\sin (\omega_-+x) = \frac{1}{2} \cot \left(\frac{1}{2}\right) \sin x -\frac{1}{2} \cos x Particularly, : \operatorname{st}\sin \omega_-=-1/2 , : \operatorname{st}\sin \omega_+=1/2 , : \operatorname{st}\sin \tau=0 : \operatorname{st}\cos (\omega_-+x) = \frac{1}{2} \csc \left(\frac{1}{2}\right) \cos \left(\frac{1}{2}- x \right) : \operatorname{st}\sin (a\omega_-+x) = \frac{a}{2} \cot \left(\frac{a}{2}\right) \sin x -\frac{a}{2} \cos x : \operatorname{st}\cos (a\omega_-+x) = \frac{a}{2} \csc \left(\frac{a}{2}\right) \cos \left(\frac{a}{2}- x\right) : \operatorname{st}\cos (\pi\tau+x)=-\frac\pi{2}\cos x : \operatorname{st}\sin (\pi\tau+x)=-\frac\pi{2}\sin x : \operatorname{st}\frac{(\omega_-+x)^{\omega_-+x}}{\omega_-^{\omega_-}}=\Gamma(x+1) Relations between standard parts of trigonometric and inverse trigonometric functions : \operatorname{st}\left(1-\cosh \left(2 x \omega _+\right)\right)=\operatorname{st}\frac{x}{\pi} \operatorname{arctanh}\left(\frac{x}{\pi\omega _+}\right)=\operatorname{st}\frac{x}{\pi} \operatorname{arccoth}\left(\frac{\pi \omega _+}{x}\right)=1-x \coth (x) : \operatorname{st}\frac{z}{2\pi }\ln \left(\frac{\omega _+-\frac{z}{2\pi }}{\omega _-+\frac{z}{2\pi }}\right)=\operatorname{st} \cos (z\omega_-)=\operatorname{st} \cos (z\omega_+)=\frac z2 \cot \left(\frac z2\right) Other identities : \sum_{k=0}^\infty 1=\omega_+ : \sum_{k=1}^\infty 1=\omega_- : \sum_{k=0}^\infty (-1)^k=\frac12 : \int_0^\infty dx = \omega_-+1/2=\tau : \int_0^\infty x\, dx =\sum_{k=0}^\infty k-\frac1{24} : \int_0^\infty \cos x\, dx =0 : \int_0^\infty \sin x\, dx =1 : \delta(0)=\frac{2\tau}\pi